# A pizza place offers 3 different cheeses and 8 different toppings. In how many ways can a pizza be made with 1 cheese and 3 toppings?

Mar 6, 2018

We assume the three toppings must be different.

#### Explanation:

Cheese : $3$ choices

Toppings : $8$ choices for the first, $7$ for the second and $6$ for the third, a total of $8 \times 7 \times 6 = 336$, IF the order of toppings were important -- which it isn't. This number is called the number of permutations .

Three things can be ordered in 6 ways (try this), so in the 336 permutations, there are groups of 6 that amount to the same combination :
123=132=213=231=312=321, etc.

So we have to divide the number of permutations by the number of orders to reach the number of combinations:
There are thus 336 : 6 = 56 possibilities for the toppings.

Since we need cheese AND toppings we multiply:

Number of different pizzas: 3 x 56 = 168.

Calculator : if you have the nCr function the answer would be:
3 x 8 nCr 3 = 168